L(s) = 1 | + 1.12i·3-s − 0.0952i·7-s + 1.72·9-s + 6.35·13-s − 4.53·17-s + 4.35i·19-s + 0.107·21-s − 9.35i·23-s + 5·25-s + 5.33i·27-s − 4.09·29-s + 8.71·37-s + 7.16i·39-s + 6i·47-s + 6.99·49-s + ⋯ |
L(s) = 1 | + 0.651i·3-s − 0.0360i·7-s + 0.575·9-s + 1.76·13-s − 1.10·17-s + 0.999i·19-s + 0.0234·21-s − 1.95i·23-s + 25-s + 1.02i·27-s − 0.760·29-s + 1.43·37-s + 1.14i·39-s + 0.875i·47-s + 0.998·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.847190321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.847190321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 3 | \( 1 - 1.12iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 0.0952iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.35T + 13T^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 23 | \( 1 + 9.35iT - 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.71T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 16.0iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.944860121109089065726380780328, −8.841981572187319847586600192375, −8.558832372606625280775071292550, −7.31767527618999705664466064113, −6.43618783404534822832572486524, −5.67664823808770301655670714003, −4.32610679317451350393578730471, −4.04809674069850140358972213701, −2.68627560934883496493254520034, −1.21310877272745023636475915826,
1.00442505296506674979221834573, 2.09017692081194110916467871165, 3.45682292045642817384465841761, 4.37228381782609191746803685056, 5.53888091236379014148721565917, 6.45194329261096817876321663820, 7.08925819519704063987292632205, 7.925327301794570934130660008428, 8.862480050657730083160453033771, 9.420590300011157922780530716071